Let us begin with a short informal discussion on intrinsic scaling.
The proof of interior Hölder continuity for non-degenerate parabolic equations relies on estimates of the oscillation of the solution \(u\)  in a family of shrinking space-time cylinders \(\lbrace Q_n\rbrace _{n=0}^\infty \) , see [1]}.
These cylinders are roughly of the form \(Q_n=B_{R_n}(x_0) \times (-R_n^2+t_0,t_0)\)  and their space-time scaling reflects the structure of the equation.
For instance, the heat equation \(u_t=\Delta u\)  is scale invariant under coordinate transformations of the type \((x,t)\mapsto (\lambda x,\lambda ^2 t)\) , \(\lambda \in {\mathbb {R}}\) , that is, if \(u\)  is a solution of the equation then \(u_\lambda \)  given by \(u_\lambda (x,t)=u(\lambda x,\lambda ^2,t)\)  is a solution as well.
The method of shrinking cylinders with this parabolic scaling fails to hold for degenerate equations such as the porous medium equation \(u_t=\Delta u^m\) .
However, we can rewrite this equation as
\(\frac{1}{m} \frac{1}{u^{m-1}}(u^m)_t=\Delta u^m,\) 
